Marin van Heel

Angular reconstitution: A posteriori assignment of projection directions for 3D reconstruction

Abstract In computerized tomography as well as in most problems of three-dimensional reconstruction from projections, one knows from the experimental set-up the angular relationships between the projections from which the reconstruction is to be calculated. A serious difficulty is encountered when the angles are not known. In this paper, a method of “angular reconstitution” is described, which allows the a posteriori determination of the relative angular orientations of the projections and thus enables the three-dimensional reconstruction of the object to be calculated. For asymmetric objects, a minimum of three projections is required, which should not be related by a tilt around a single rotation axis. The method can be applied to determine the three-dimensional structure of biological macromolecules based on electron micrographs of randomly oriented individual molecules. Angular reconstitution, in combination with multivariate statistical techniques to classify and average the characteristic views of a molecule forms a complete, self-contained methodology for molecular structure analysis by electron microscopy.

The 70S Escherichia coli ribosome at 23 å resolution: fitting the ribosomal RNA

BACKGROUNDnThe ribosome--essential for protein synthesis in all organisms--has been an evasive target for structural studies. The best available structures for the 70S Escherichia coli ribosome or its 30S and 50S subunits are based on electron microscopical tilt experiments and are limited in resolution to 28-55 A. The angular reconstitution approach, which exploits the random orientations of particles within a vitreous ice matrix, can be used in conjunction with cryo-electron microscopy to yield a higher-resolution structure.nnnRESULTSnOur 23 A resolution map of the 70S ribosome elucidates many structural details, such as an extensive system of channels within the 50S subunit and an intersubunit gap ideally shaped to accommodate two transfer RNA molecules. The resolution achieved is sufficient to allow the preliminary fitting of double-helical regions of an earlier three-dimensional ribosomal RNA model.nnnCONCLUSIONSnAlthough we are still a long way from attaining an atomic-resolution structure of the ribosome, cryo-electron microscopy, in combination with angular reconstitution, is likely to yield three-dimensional maps with gradually increasing resolution. As exemplified by our current 23 A reconstruction, these maps will lead to progressive refinement of models of the ribosomal RNA.

Correlation functions revisited

Abstract Correlation functions (auto-correlations, cross-correlations, rotational correlations) are used extensively in general signal processing, in Fourier optics, and particularly in electron microscopical image processing. In spite of their widespread and successful use in a broad spectrum of pattern-matching procedures, we believe there is reason to reconsider their general use. Correlation functions have the intrinsic limitation that they are calculated by squaring or multiplication operations in Fourier space. This fact causes the strong frequency components in the data (typically the low-frequency ones) to become overwhelmingly stronger relative to the weaker components, typically the high-frequency components associated with the fine details in the data in which we are especially interested. By dividing Fourier components by the square roots of their amplitudes, these unfavorable effects can be avoided resulting in substantially better correlation functions. Our new mutual-correlation function (MCF) and self-correlation function (SCF), which relate to each other the way the conventional cross-correlation function (CCF) relates to the auto-correlation function (ACF), can supersede conventional correlation functions for most purposes. We have subjected the novel correlation functions to practical tests in routine image-alignment procedures and found the results satisfactory. The problems we experienced with conventional squared correlation functions are likely to hinder triple-correlation function applications to an even greater extent.

Invariant classification of molecular views in electron micrographs

Biological macromolecules can exhibit many different orientations in electron microscopical preparations. In particular in vitreous-ice-embedded specimens, the number of different views can be high. Existing techniques of analysis require the alignment of the molecular views relative to one or more reference images with cross-correlation (matched filtering) techniques and are somewhat unsatisfactory because of the high noise level and the large number of different views in such images. We here propose a method in which first rotation-, translation- and mirror-invariant functions are derived from the large set of input images. These functions are subsequently classified automatically using multivariate statistical classification techniques. The different molecular views in the images can therewith be found without bias, provided that a statistically significant number of copies of the views are present in the data set. The basic ideas are exemplified with realistic model data.

A new family of powerful multivariate statistical sequence analysis techniques

Abstract A novel multivariate statistical approach is presented for extracting and exploiting intrinsic information present in our ever-growing sequence data banks. The information extraction from the sequences avoids the pitfalls of intersequence alignment by analyzing secondary invariant functions derived from the sequences in the data bank rather than the sequences themselves. Such typical invariant function is a 20 × 20 histogram of occurrences of amino acid pairs in a given sequence or fragment thereof. To illustrate the potential of the approach an analysis of 10,000 protein sequences from the National Biomedical Research Foundation Protein Identification Resource is presented, whose analysis already reveals great biological detail. For example, ζ-hemoglobin is found to lie close to amphibian and fish α-hemoglobin which, in turn, is an important clue to the physiological function of this mammalian early embryonic hemoglobin. The multivariate statistical framework presented unifies such apparently unrelated issues as phylogenetic comparisons between a set of sequences and distance matrices between the constituents of the biological sequences. The Multivariate Statistical Sequence Analysis (MSSA) principles can be used for a wide spectrum of sequence analysis problems such as: assignment of family memberships to new sequences, validation of new incoming sequences to be entered into the database, prediction of structure from sequence, discrimination of coding from non-coding DNA regions, and automatic generation of an atlas of protein or DNA sequences. The MSSA techniques represent a self-contained approach to learning continuously and automatically from the growing stream of new sequences. The MSSA approach is particularly likely to play a significant role in major sequencing efforts such as the human genome project.

Classification of image data in conjugate representation spaces

Over the past few years, automatic multivariate statistical classification techniques have successfully been used for analyzing large and noisy electron-microscopic data sets. After the raw data are compressed with eigenvector–eigenvalue procedures, classes of images are formed, using unsupervised clustering procedures. The classes elucidate even subtle differences existing within the data set. Here we extend these methods to find classes of pixels or features in the images (or other n-dimensional signals) that exhibit a homogeneous statistical behavior throughout the data set. This feature extraction—itself a form of data compression—is mathematically entirely symmetric to the determination of the image classes and also serves the purpose of revealing the information present in the set of input images. The properties of simultaneous representations of the image-space and feature-space data onto the same two-dimensional map are discussed in relation to the metrics used in both spaces. Model data are used to illustrate the basic ideas.

Stacked bilayer helices: a new structural organization of amphiphilic molecules

Intriguing helical fibres can be created by self-assembly of simple chiral amphiphilic molecules. We study the parameters governing this spontaneous self-organization by three-dimensional (3D) electron microscopy of the helical fibres embedded in a vitreous ice-matrix. Different stable helices are generated reproducibly using specific combinations of the control parameters in our system. All fibres with diameters less than 25 nm consist of a narrow stack of compartmented bilayers twisted into a left-handed helix. Our novel helical 3D reconstruction procedures in combination with specialized cryomicroscopical specimen preparation, can rapidly elucidate the structure of such helical assemblies. This approach may complement or even replace existing diffraction-based methodologies.

Quanternary structure of multihexameric arthropod hemocyanins

Arthropod hemocyanins are large oligomeric oxygen-transporting proteins with molecular weight ranging from 450 kDa in the spiny lobster (Panulirus interruptus) up to more than 3.6 mDa in the horseshoe crab (Limulus polyphemus). Hemocyanins from different species consist of one or multiple copies of a hexameric building block (of 450 kDa) and are sufficiently large to be easily visualized in the electron microscope. Arthropod hemocyanins were among the first macromolecules studied by multivariate statistical image analysis techniques. We present an overview of the different characteristic molecular images of various multihexameric (1 × 6, 2 × 6, 4 × 6, and 8 × 6) assemblies as these occur in electron-microscopical preparations. We also model the different assemblies in three dimensions by merging multiple copies of the X-ray-diffraction electron density of the single hexameric hemocyanin of Panulirus interruptus. By making correct enantiomeric decisions while merging the densities at the various levels of assembly and by fine-tuning the assembly parameters used, a good match can be obtained between the microscopical images and two-dimensional projections calculated from the three-dimensional (3D) model densities. Knowledge of the quaternary structures of this intricate hierarchical family of oligomers is essential for understanding the allosteric interactions associated with their strong oxygen-binding cooperativity.

Three-dimensional reconstruction of a human metaphase chromosome from electron micrographs

A complete human metaphase chromosome has been reconstructed from a series of electron microscopical projections obtained by tilting the specimen stage at 3 degree intervals from −60 to +60 degrees. The reconstructed structure is about 3.0 μm long, 1.6 μm wide, and 0.8 μm thick. The mass distribution was fairly homogeneous within the chromatids and neither a hollow nor a dense core was observed. The distribution and course of fibers observed are most consistent with a looping model of chromosome structure.

Characteristic views of prokaryotic 50S ribosomal subunits

SummaryMultivariate statistical analysis and classification techniques are powerful tools in sorting noisy electron micrographs of single particles according to their principal features, enabling one to form average images with an enhanced signal-to-noise ratio and a better reproducible resolution. We apply this methodology here to determining the characteristic views of the large (50S) ribosomal subunits from the eubacteriumEscherichia coli and the archaebacteriaMethanococcus vannielii, Sulfolobus solfataricus, andHalobacterium marismortui. Average images were obtained of the subunit in the common crown and kidney projections, but views of the particle in orientations intermediate between these two extremes were also elucidated for all species. These averages show reproducible detail of up to 2.0 nm resolution, thus enabling the visualization and interspecies comparison of many structural features as a first step toward comparing the actual three-dimensional structures. Our results disprove evolutionary lineages recently postulated on the basis of electron microscopical images of ribosomal subunits.